| L(s) = 1 | + 3-s + 2.08·5-s + 5.03·7-s + 9-s − 0.828·11-s − 2.94·13-s + 2.08·15-s + 4.82·17-s + 2.82·19-s + 5.03·21-s − 4.16·23-s − 0.656·25-s + 27-s + 7.97·29-s − 5.03·31-s − 0.828·33-s + 10.4·35-s − 7.11·37-s − 2.94·39-s + 8.82·41-s − 12.4·43-s + 2.08·45-s − 4.16·47-s + 18.3·49-s + 4.82·51-s − 12.1·53-s − 1.72·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.932·5-s + 1.90·7-s + 0.333·9-s − 0.249·11-s − 0.817·13-s + 0.538·15-s + 1.17·17-s + 0.648·19-s + 1.09·21-s − 0.869·23-s − 0.131·25-s + 0.192·27-s + 1.48·29-s − 0.903·31-s − 0.144·33-s + 1.77·35-s − 1.16·37-s − 0.471·39-s + 1.37·41-s − 1.90·43-s + 0.310·45-s − 0.607·47-s + 2.61·49-s + 0.676·51-s − 1.66·53-s − 0.232·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.015883086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.015883086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 - 2.08T + 5T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 2.94T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4.16T + 23T^{2} \) |
| 29 | \( 1 - 7.97T + 29T^{2} \) |
| 31 | \( 1 + 5.03T + 31T^{2} \) |
| 37 | \( 1 + 7.11T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 + 4.16T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 + 7.11T + 61T^{2} \) |
| 67 | \( 1 + 2.34T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 5.03T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.526937356717892454530632849582, −8.541366920595151993656009855334, −7.87793360777142888984109836720, −7.36829017285910341646468170176, −6.06110014603452595243852196163, −5.16258685120708637575891017186, −4.66617330088337856098167598277, −3.29058823319757676057720587674, −2.12180394044258468573790142845, −1.44835202152548496308962733591,
1.44835202152548496308962733591, 2.12180394044258468573790142845, 3.29058823319757676057720587674, 4.66617330088337856098167598277, 5.16258685120708637575891017186, 6.06110014603452595243852196163, 7.36829017285910341646468170176, 7.87793360777142888984109836720, 8.541366920595151993656009855334, 9.526937356717892454530632849582
